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As the title suggests, the larger gaps are to be filled with Kumiko. The smaller gaps will be left empty. I though about a simple pattern as in the demo lantern on the right in this picture.
Do you have other ideas? All I can currently do with my mitre saw are square patterns. Cannot do triangles, unfortunately.
https://www.youtube.com/watch?v=kxG-NbUzMVE
Awesome! Unfortunately, that's exactly the kind of triangle pattern my mitre saw is too inaccurate for.
Make yourself a shooting board at the right angle.
If I understand you correctly, that wouldn't help with the 30° cuts into the strips that make up the frame. Perhaps I need to upgrade my jig for that, but I haven't yet found a solid and easy approach to that.
https://youtu.be/JbpwDufvzSo
This is a 90 degree shooting board. You can make one where the fence is 30 degrees and use it to ensure your angles. It's a bit slow, but so is kumiko in general.